Assuming black box access to the oracles \(U_A\)  and \(U_{A^\perp }\) , it was proved in [1]} that a counterfeiter needs at least \(\Omega (\sqrt{\epsilon } 2^{n / 4})\)  queries to prepare a state \(\rho \)  such that \(\bra {A}^{\otimes 2} \rho \ket {A}^{\otimes 2} \ge \epsilon \) . However, there has been no provably secure instantiation of these oracles to this date.
